Let be a compact connected subset of the complex plane that contains more than one point. Show that the family of meromorphic functions on a domain that omits (that is, with range in is a normal family of meromorphic functions.
The hint in the back of the book says to map conformally onto , apply thesis version of Montel's theorem. That version states:
Suppose is a family of analytic functions on a domain such that is uniformly bounded on each compact subset of . Then every sequence in has a subsequence that converges normally in , that is, uniformly on each compact subset of .
I am still confused on how to prove this. I would appreciate some help on this problem. Thanks.